Abelian varieties are de Rham $K(π,1)$
Vo Quoc Bao, Quang-Khai Nguyen
TL;DR
The paper develops and applies the de Rham K(π,1) framework by introducing δ^i_{X/k}(V) as a bridge between group-scheme cohomology of the differential fundamental group π^diff(X/k) and de Rham cohomology. It proves that abelian varieties over characteristic 0 are de Rham K(π,1), using a transcendental-to-algebraic strategy that leverages base change, the Lyndon–Hochschild–Serre spectral sequence, and the Albanese morphism. It then analyzes the cohomology of the abelianization π^diff(X/k)^{ab}, showing finite-dimensionality and vanishing beyond 2g for abelian X of dimension g, and establishes a canonical isomorphism with the Alb_X side via η^i(V). The results draw strong parallels with topological K(π,1) phenomena and provide a cohesive bridge between de Rham theory, Tannakian formalism, and Albanese-type universal objects, with explicit consequences for Euler characteristics and cohomology comparisons both in the algebraic and topological settings.
Abstract
Motivated by the work of Esnault-Hai, one has the notion of de Rham $K(π,1)$ schemes, defined as follows. Given a smooth proper geometrically connected scheme $X$ over a field $k$ of characteristic 0 and a base point $x \in X(k)$, one can define its differential fundamental group $π^{\mathrm{diff}}(X/k)$, which comes from the Tannakian duality of the category of coherent integrable connections on $X$. Using the formalism of $δ$-functor, one can define natural morphisms between the group-scheme cohomology of $π^{\mathrm{diff}}(X/k)$ and the de Rham cohomology of $X$. One says that $X$ with $x \in X(k)$ is de Rham $K(π,1)$ if such morphisms are all isomorphisms. In this article, we prove that abelian varieties in characteristic 0 are de Rham $K(π,1)$. In the second part of the article, we study the group-scheme cohomology of the abelianization of the differential fundamental group of a smooth proper geometrically connected scheme via its Albanese variety.